A159854 → basis A013609 → D

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence

\(\left(3 + 2\,x + n + 2\,n x\right)\,P_{n}(x) = \left(8 + 16\,x + 8\,x^{2} + 2\,n + 6\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 4 + 8\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	4	8															12
1	10	32	24														66
2	24	104	144	64													336
3	56	304	600	512	160												1632
4	128	832	2112	2624	1600	384											7680
5	288	2176	6720	10880	9760	4608	896										35328
6	640	5504	19968	39680	46720	32640	12544	2048									159744
7	1408	13568	56448	132608	192640	177408	101248	32768	4608								712704
8	3072	32768	153600	415744	716800	817152	616448	296960	82944	10240							3145728
9	6656	77824	405504	1241088	2472960	3354624	3139584	2002944	834048	204800	22528						13762560
10	14336	182272	1044480	3563520	8048640	12644352	14106624	11182080	6174720	2263040	495616	49152					59768832
11	30720	421888	2635776	9912320	25006080	44605440	57716736	54607872	37509120	18247680	5969920	1179648	106496				257949696
12	65536	966656	6537216	26853376	74792960	149225472	219512832	241139712	197885952	119848960	52084736	15384576	2768896	229376			1107296256
13	139264	2195456	15974400	71139328	216719360	477954048	787218432	984023040	938336256	679444480	367837184	144408576	38871040	6422528	491520		4731174912
14	294912	4947968	38535168	184877056	611287040	1476034560	2689662976	3767402496	4090724352	3444080640	2230452224	1091371008	390627328	96567296	14745600	1048576	20132659200

Technical details

Node id: n-5ce2a3f3aa7d1655 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-645712c1d17a2bff

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:    4,     8;
  1:   10,    32,     24;
  2:   24,   104,    144,      64;
  3:   56,   304,    600,     512,     160;
  4:  128,   832,   2112,    2624,    1600,     384;
  5:  288,  2176,   6720,   10880,    9760,    4608,     896;
  6:  640,  5504,  19968,   39680,   46720,   32640,   12544,    2048;
  7: 1408, 13568,  56448,  132608,  192640,  177408,  101248,   32768,   4608;
  8: 3072, 32768, 153600,  415744,  716800,  817152,  616448,  296960,  82944,  10240;
  9: 6656, 77824, 405504, 1241088, 2472960, 3354624, 3139584, 2002944, 834048, 204800, 22528;

%F Formula

The row polynomials satisfy (n+3+(2*n+2)*x)*P(n,x) = (2*n+8+(6*n+16)*x+(4*n+8)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 4 + 8*x.

%t Mathematica

P[0, x_] := 4 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 8 + (6*n + 16)*x + (4*n + 8)*x^2)*P[n - 1, x])/(n + 3 + (2*n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 4 + 8*x; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 8 + (6*n + 16)*x + (4*n + 8)*x^2)*v[i - 1])/(n + 3 + (2*n + 2)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of Riordan array (1-x,x/(1-x)).?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Riordan array (1-x,x/(1-x)).?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

A391533 → reverse → D

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.